page_title: Normally distributed and uncorrelated does not imply independent
section_title: A symmetric example
hierarchical_section_title: Normally distributed and uncorrelated does not imply independent / Examples / A symmetric example
caption_attribution_description: English: Two normally distributed, uncorrelated but dependent variables.
context_page_description: In probability theory, although simple examples illustrate that linear uncorrelatedness of two random variables does not in general imply their independence, it is sometimes mistakenly thought that it does imply that when the two random variables are normally distributed.
  To say that the pair of random variables has a bivariate normal distribution means that every linear combination of and for constant coefficients and has a univariate normal distribution. In that case, if and are uncorrelated then they are independent. However, it is possible for two random variables and to be so distributed jointly that each one alone is marginally normally distributed, and they are uncorrelated, but they are not independent; examples are given below.
context_section_description: Suppose  has a normal distribution with expected value 0 and variance 1. Let  have the Rademacher distribution, so that  or , each with probability 1/2, and assume  is independent of .  Let . Then 
   and  are uncorrelated;
  both have the same normal distribution; and
   and  are not independent.
  To see that  and  are uncorrelated, one may consider the covariance : by definition, it is
  Then by definition of the random variables , , and , and the independence of  from , one has
  To see that  has the same normal distribution as , consider
  (since  and  both have the same normal distribution), where  is the cumulative distribution function of the normal distribution..
  To see that  and  are not independent, observe that  or that .
  Finally, the distribution of the simple linear combination  concentrates positive probability at 0: .  Therefore, the random variable  is not normally distributed, and so also  and  are not jointly normally distributed (by the definition above).
caption: Joint range of 
    
      
        
          X
        
      
      {\displaystyle X}
    
   and 
    
      
        
          Y
        
      
      {\displaystyle Y}
    
  . Darker indicates higher value of the density function.
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